(i),(ii)\(\Rightarrow \) (iii) By [1]} and [2]}, \(\mbox{depth}\,_R(M\otimes _RN)=\mbox{depth}\,_R(N)-\mbox{CI-dim}\,_R(M)\) . As \(\mbox{Ext}\,^i_R(\mbox{Tr}\,M,N)=0\)  for all \(i>0\) , \(M\otimes _RN\cong \mbox{Hom}\,_R(M^*,N)\)  by Theorem REF 
and also \(\mbox{Ext}\,^i_R(M^*,N)=0\)  for all \(i>0\) , by the exact sequence (REF ). Therefore, \(\mbox{depth}\,_R(\mbox{Hom}\,_R(M^*,N))=\mbox{depth}\,_R(N)\)  by [1]}. Hence \(\mbox{CI-dim}\,_R(M)=0\) .
(ii),(iii)\(\Rightarrow \) (i) Set \(K=\mbox{Tr}\,M\)  and \(c=\mbox{cx}\,_R(M)\) . By Lemma REF , \(\mbox{CI-dim}\,_R(K)=0\) . By Theorem REF , \(\mbox{Tor}\,_1^R(\mathcal {T}_{i+1}K,N)=0\)  for all \(i>0\) . As \(\mbox{Ext}\,^i_R(K,R)=0\)  for all \(i>0\) , \(\underline{\mathcal {T}_iK}\cong \underline{\Omega \mathcal {T}_{i+1}K}\)  for all \(i>0\)  and so
\(\mbox{Tor}\,_i^R(\mathcal {T}_jK,N)=0\)  for all \(j>1\)  and \(1\le i<j\) . As \(\underline{M}\cong \underline{\mbox{Tr}\,K}\) , \(\mbox{cx}\,_R(\mbox{Tr}\,K)=c\)  and so \(\mbox{cx}\,_R(\mathcal {T}_iK)=c\)  for all \(i>0\) . Since \(\mbox{Tor}\,_i^R(\mathcal {T}_{c+2}K,N)=0\)  for \(1\le i\le c+1\)  and \(\mbox{CI-dim}\,_R(\mathcal {T}_{c+2}K)=0\) , then \(\mbox{Tor}\,_i^R(\mathcal {T}_{c+2}K,N)=0\)  for all \(i>0\) , by[4]}. Therefore, \(\mbox{Tor}\,_i^R(\mathcal {T}_1K,N)=0\)  for all \(i>0\) . As \(\underline{M}\cong \underline{\mbox{Tr}\,K}\) , \(\mbox{Tor}\,^i_R(M,N)=0\)  for all \(i>0\) .
(i), (iii)\(\Rightarrow \) (ii) By [2]}\(, \mbox{G-dim}\,_R(M)=\mbox{CI-dim}\,_R(M)=0\)  and by [6]}, \(M\)  has reducible complexity. Therefore, \(\mbox{Ext}\,^i_R(\mbox{Tr}\,M,N)=0\)  for all \(i>0\)  by Lemma REF .
